A Statistical Geometry Approach to the Study of Protein Structure

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A Statistical Geometry Approach to the Study of
Protein Structure

• Majid Masso
• Bioinformatics and Computational Biology
• George Mason University

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Protein Basics

• formed by linearly linking amino acid residues
  (aas are the building blocks of proteins)
• 20 distinct aa types
• A,C,D,E,F,G,H,I,K,L,M,N,P,Q,R,S,T,V,W,Y

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Protein Basics

• genes code, or blueprint
• proteins product, or building
• protein structure gives rise to function
• why do things go wrong?
• mistakes in blueprint
• incorrectly built, or nonexistent buildings
• Protein Data Bank (PDB) repository of protein
  structural data, including 3D coords. of all
  atoms (www.rcsb.org/pdb/)

PDB ID 1REZ Structure reference Muraki M.,
Harata K., Sugita N., Sato K., Origin of
carbohydrate recognition specificity of human
lysozyme revealed by affinity labeling,
Biochemistry 35 (1996)
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Computational Geometry Approach to Protein
Structure Prediction

• Tessellation
• protein structure represented as a set of points
  in 3D, using Ca coordinates
• Voronoi tessellation convex polyhedra, each
  contains one Ca , all interior points closer to
  this Ca than any other
• Delaunay tessellation connect four Ca whose
  Voronoi polyhedra meet at a common vertex
• vertices of Delaunay simplices objectively define
  a set of four nearest-neighbor residues
  (quadruplets)
• 5 classes of Delaunay simplices
• Quickhull algorithm (qhull program), Barber et
  al., UMN Geometry Center

Voronoi/Delaunay tessellation in 2D space.
Voronoi tessellation-dashed line, Delaunay
tessellation-solid line (Adapted from Singh R.K.,
et al. J. Comput. Biol., 1996, 3, 213-222.)
Five classes of Delaunay simplices. (Adapted from
Singh R.K., et al. J. Comput. Biol., 1996, 3,
213-222.)
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Counting Quadruplets

• assuming order independence among residues
  comprising Delaunay simplices, the maximum number
  of all possible combinations of quadruplets
  forming such simplices is 8855

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Residue Environment Scores

• log-likelihood
• normalized frequency of quadruplets
  containing residues i,j,k,l in a representative
  training set of high-resolution protein
  structures with low primary sequence identity
• i.e., total number of quadruplets in
  dataset containing only residues i,j,k,l divided
  by total number of observed quadruplets
• frequency of random occurrence of the
  quadruplet (multinomial)
• i.e.,
• total number of occurrences of residue i
  divided by total number of residues in the
  dataset
• , where n number of distinct
  residue types in the
• quadruplet, and t i is the
  number of residues of type i.

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Residue Environment Scores

• total statistical potential (topological score)
  of protein sum the log-likelihoods of all
  quadruplets forming the Delaunay simplices
• individual residue potentials sum the
  log-likelihoods of all quadruplets in which the
  residue participates (yields a 3D-1D potential
  profile)

PDB ID 3phvHIV-1 Protease Monomer 99 amino
acids (total potential 27.93)
Structure reference R. Lapatto, T. Blundell, A.
Hemmings, et al., X-ray analysis of HIV-1
proteinase at 2.7 Å resolution confirms
structural homology among retroviral enzymes,
Nature 342 (1989) 299-302.
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HIV-1 Protease Comprehensive Mutational Profile
(CMP)

• mutate 19 times the residue present at each of
  the 99 positions in the primary sequence
• get total potential and potential profile of each
  artificially created mutant protein
• create 20x99 matrix containing total potentials
  of all the single residue mutants
• columns labeled with residues in the primary
  sequence of wild-type (WT) HIV-1 protease
  monomer, and rows labeled with the 20 naturally
  occurring amino acids
• subtract WT total potential (TP) from each cell,
  then average columns to get CMP
• CMPj (mutant TP)ij-(WT TP)
  (mutant TP)ij-27.93 , j1,,99

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Structure-Function Correlations

• 536 single point missense mutations
• 336 published mutants Loeb D.D., Swanstrom R.,
  Everitt L., Manchester M., Stamper S.E.,
  Hutchison III C.A. Complete mutagenesis of the
  HIV-1 protease. Nature, 1989, 340, 397-400
• 200 mutants provided by R. Swanstrom (UNC)
• each mutant placed in one of 3 phenotypic
  categories, positive, negative, or intermediate,
  based on activity
• mutant activity compared with change in
  sequence-structure compatibility elucidated by
  potential data

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Observations

• set of mutants with unaffected protease activity
  exhibit minimal (negative) change in potential
• set of mutants that inactivate protease exhibit
  large negative change in potential, weighted
  heavily by NC
• set of mutants with intermediate phenotypes
  exhibit moderate negative change in potential
  (similar among C and NC) wide range for
  intermediate phenotype in the experiments

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Acknowledgements

• Iosif Vaisman (Ph.D. advisor, first to apply
  Delaunay to protein structure)
• Zhibin Lu (Java programs for calculating
  statistical potentials from tessellations)
• Ronald Swanstrom (experimental HIV-1 protease
  mutants and activity measure)